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Gu, Ying-Qiu. "Clifford Algebras, Hypercomplex Numbers and Nonlinear Equations in Physics". Geometry, Integrability and Quantization 25 (2023): 47–72. http://dx.doi.org/10.7546/giq-25-2023-47-72.
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Hypercomplex number systems are vector algebras with the definition of multiplication and division of vectors, satisfying the associativity and distributive law. In this paper, some new types of hypercomplex numbers and their fundamental properties are introduced, the Clifford algebra formalisms of hydrodynamics and gauge field equations are established, and some novel consistent conditions helpful to understand the properties of solutions to nonlinear physical equations are derived. The coordinate transformation and covariant derivatives of hypercomplex numbers are also discussed. The basis elements of the hypercomplex numbers have group-like properties and satisfy a structure equation $\A^2=n\A$. The hypercomplex number system integrates the advantages of algebra, geometry and analysis, and provides a unified, standard and elegant language and tool for scientific theories and engineering technology, so it is easy to learn and use. The description of mathematical, physical and engineering problems by hypercomplex numbers is of neat formalism, symmetric structure and standard derivation, which is especially suitable for the efficient processing of the higher dimensional complicated systems.
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Kalinovsky, Ya A., Yu E. Boyarinova, Ya V. Khitsko i A. S. Sukalo. "Use of Methods for Generating Isomorphic Hypercomplex Number Systems to Increase the Efficiency of Multiplying Hypercomplex Numbers". Èlektronnoe modelirovanie 40, nr 5 (25.10.2018): 27–40. http://dx.doi.org/10.15407/emodel.40.05.027.
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Кalinovskiy, Ya А., i Yu E. Boiarinova. "Method for Representing an Exponent in a Fifth-dimensional Hypercomplex Number Systems Using a Hypercomplex Computing Software". Èlektronnoe modelirovanie 43, nr 6 (6.12.2021): 3–18. http://dx.doi.org/10.15407/emodel.43.06.003.
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The structure of method for constructing a representation of an exponential function in hypercomplex number systems (HNS) by the method of solving an associated system of linear differential equations is considered. Brief information about the hypercomplex computing software (HCS) is given. With the use of HCS, the necessary cumbersome operations on symbolic expressions were performed when constructing the representation of the exponent in the fifthdimensional HNS. Fragments of programs in the environment of HCS and results of symbolic calculations are resulted.
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Hauser, Jochem, i Walter Dröscher. "Gravity Beyond Einstein? Part II: Fundamental Physical Principles, Number Systems, Novel Groups, Dark Energy, and Dark Matter, MOND". Zeitschrift für Naturforschung A 74, nr 5 (27.05.2019): 387–446. http://dx.doi.org/10.1515/zna-2018-0559.
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AbstractThis article attempts to explain the underlying physics of several recent experiments and astrophysical observations that have been mystifying the physics community for quite some time. So far, none of the advanced theories beyond the standard models of particle physics and cosmology have shown sufficient potential to resolve these mysteries. The reason for this failure may lie in the fact that these theories are based on the concept of extra space dimensions that appears to be in conflict with numerous experiments, in particular with recent Large Hadron Collider data. Therefore, the novel idea of extra number systems is introduced, replacing the idea of extra space dimensions. This approach is complemented by a set of fundamental physical principles that provide the constraints and guidelines for a modified physical formulation in agreement with known experimental reality. However, such a theory requires novel physical concepts in conjunction with novel symmetry groups. These groups give rise to additional types of matter, termed hypercomplex masses (which are responsible for the extreme hypercomplex gravitational fields, see below, and are also denoted as matter flavour), including, for instance, particles of negative mass, identified with dark matter. Furthermore, four-dimensional Minkowski spacetime, assumed to be a quasi de Sitter space $dS^{1,3}$dual spacetime, $DdS^{1,3}$, with imaginary time coordinate; that is, time is a complex quantity. The three spatial coordinates are shared by the two spacetimes. Dark matter is assumed to reside in $DdS^{1,3}$ and therefore is principally invisible. On the other hand, its gravitational interaction with ordinary matter (m ≥ 0) in spacetime $dS^{1,3}$ is directly perceptible. The novel group structure predicts the existence of a fourth particle family of negative masses; that is, besides the dark matter particle χ of mass $m_{\chi}\approx-80.77$ GeV/c2, there is the dark neutrino νχ of mass $m_{\nu_{\chi}}\approx-3.23$ eV/c2. Moreover, the hypercomplex group structure of gravity ($SU(2)\times SU(2)$) postulates three gravitational bosons for cosmological fields [resulting from Einstein’s theory of general relativity (GR)], the graviton $\nu_{G_{N}}$ with spin 2, the novel gravitophoton $\nu_{gp}$ with spin 1 (existence of weak gravitomagnetic fields of GR), and the quintessence particle νq with spin 0, which, when present, mediates an interaction between ordinary matter (m ≥ 0) and the ubiquitous scalar field of dark energy. In addition, the existence of extreme gravity fields (hypercomplex gravity) is postulated, based on the second group SU(2), and an interaction between electromagnetism and hypercomplex gravity is predicted, mediated by three additional hypercomplex-gravity bosons. Some long-standing problems of cosmology will be addressed; namely, the Big Bang scenario and the origin of dark energy and the nature of dark matter and their relation to the modified Newtonian dynamics hypothesis will be discussed.
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KALINOVSKY, Ya A., i Yu E. BOYARINOVA. "The Metod for Research of Isomorphism of Indecomposable Hypercomplex Number Systems." Èlektronnoe modelirovanie 39, nr 3 (6.06.2017): 61–76. http://dx.doi.org/10.15407/emodel.39.03.061.
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Wang, Xingyuan, i Tao Jin. "Hyperdimensional generalized M–J sets in hypercomplex number space". Nonlinear Dynamics 73, nr 1-2 (5.03.2013): 843–52. http://dx.doi.org/10.1007/s11071-013-0836-5.
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Cariow, Aleksandr, i Oleg Finko. "Special Issue: Real, Complex and Hypercomplex Number Systems in Data Processing and Representation". Applied Sciences 13, nr 11 (28.05.2023): 6563. http://dx.doi.org/10.3390/app13116563.
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The evolution of human society is inevitably associated with the widespread development of computer technologies and methods, and the constant evolution of the theory and practice of data processing, as well as the need to solve increasingly complex problems in computational intelligence, have inspired the use of complex and advanced mathematical methods and formalisms for representing and processing big data sets [...]
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DEMİR, SÜLEYMAN, MURAT TANIŞLI i TÜLAY TOLAN. "OCTONIC GRAVITATIONAL FIELD EQUATIONS". International Journal of Modern Physics A 28, nr 21 (20.08.2013): 1350112. http://dx.doi.org/10.1142/s0217751x13501121.
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Generalized field equations of linear gravity are formulated on the basis of octons. When compared to the other eight-component noncommutative hypercomplex number systems, it is demonstrated that associative octons with scalar, pseudoscalar, pseudovector and vector values present a convenient and capable tool to describe the Maxwell–Proca-like field equations of gravitoelectromagnetism in a compact and simple way. Introducing massive graviton and gravitomagnetic monopole terms, the generalized gravitational wave equation and Klein–Gordon equation for linear gravity are also developed.
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Klipkov, S. I. "On a new approach to the construction of hypercomplex number systems of rank two over the field of complex numbers". Ukrainian Mathematical Journal 63, nr 1 (czerwiec 2011): 158–68. http://dx.doi.org/10.1007/s11253-011-0494-z.
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Kalinovsky, J. A., Y. E. Boyarinova i J. V. Khitsko. "Method of Selecting Hypercomplex Number Systems for Modeling Digital Reversing Filters of the 3rd and 4th Orders". Èlektronnoe modelirovanie 41, nr 4 (8.08.2019): 03–18. http://dx.doi.org/10.15407/emodel.41.04.003.
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ISOKAWA, TEIJIRO, HARUHIKO NISHIMURA, NAOTAKE KAMIURA i NOBUYUKI MATSUI. "ASSOCIATIVE MEMORY IN QUATERNIONIC HOPFIELD NEURAL NETWORK". International Journal of Neural Systems 18, nr 02 (kwiecień 2008): 135–45. http://dx.doi.org/10.1142/s0129065708001440.
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Associative memory networks based on quaternionic Hopfield neural network are investigated in this paper. These networks are composed of quaternionic neurons, and input, output, threshold, and connection weights are represented in quaternions, which is a class of hypercomplex number systems. The energy function of the network and the Hebbian rule for embedding patterns are introduced. The stable states and their basins are explored for the networks with three neurons and four neurons. It is clarified that there exist at most 16 stable states, called multiplet components, as the degenerated stored patterns, and each of these states has its basin in the quaternionic networks.
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Zakarya, Mohammed, Mahmoud A. Abd-Rabo i Ghada AlNemer. "Hypercomplex Systems and Non-Gaussian Stochastic Solutions with Some Numerical Simulation of χ-Wick-Type (2 + 1)-D C-KdV Equations". Axioms 11, nr 11 (21.11.2022): 658. http://dx.doi.org/10.3390/axioms11110658.
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In this article, we discuss the (2 + 1)-D coupled Korteweg–De Vries (KdV) equations whose coefficients are variables, and stochastic (2 + 1)-D C-KdV (C-KdV) equations with the χ-Wick-type product. White noise functional solutions (WNFS) are presented with the homogeneous equilibrium principle, Hermite transform (HT), and technicality via the F-expansion procedure. By means of the direct connection between the theory of hypercomplex systems (HCS) and white noise analysis (WNA), we establish non-Gaussian white noise (NGWN) by studying stochastic partial differential equations (PDEs) with NG-parameters. So, by using the F-expansion method we present multiples of exact and stochastic families from variable coefficients of travelling wave and stochastic NG-functional solutions of (2 + 1)-D C-KdV equations. These solutions are Jacobi elliptic functions (JEF), trigonometric, and hyperbolic forms, respectively.
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Alpay, Daniel, i Ilwoo Cho. "Operators induced by certain hypercomplex systems". Opuscula Mathematica 43, nr 3 (2023): 275–333. http://dx.doi.org/10.7494/opmath.2023.43.3.275.
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In this paper, we consider a family \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\) of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations \(\{(\mathbb{C}^{2},\pi_{t})\}_{t\in\mathbb{R}}\) of the hypercomplex system \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\), and study the realizations \(\pi_{t}(h)\) of hypercomplex numbers \(h \in \mathbb{H}_{t}\), as \((2\times 2)\)-matrices acting on \(\mathbb{C}^{2}\), for an arbitrarily fixed scale \(t\in\mathbb{R}\). Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.
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Kolodezhnov, V. N. "Three-component hypercomplex numbers on two-dimensional plane". Automation and Remote Control 71, nr 6 (czerwiec 2010): 1275–82. http://dx.doi.org/10.1134/s0005117910060287.
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Zaripov, R. G. "Representations of a group of difference information functions in the extended parastatistics of non-extensive systems". Izvestiya vysshikh uchebnykh zavedenii. Fizika, nr 7 (2022): 9–15. http://dx.doi.org/10.17223/00213411/65/7/9.
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Algebraic and matrix representations of the group of information functions of the difference of nonextensive systems for three types of conformally generalized hypercomplex numbers are given. The corresponding geometries with metric functions are global Finsler geometries.
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Votiakova, Lesia, i Viktoriia Bodenchuk. "Matrix Algebra В as Euclidean Space". Mathematical and computer modelling. Series: Physical and mathematical sciences 24 (5.12.2023): 5–13. http://dx.doi.org/10.32626/2308-5878.2023-24.5-13.
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Representation of information by means of hypercomplex numerical systems is used in various problems of science and technology: in classical mechanics, solid body mechanics, electrodynamics, radio electronics, computer animation, and others [1]. Often a hypercomplex system (that is, a system whose elements are considered to be hypercomplex numbers) is understood as any finite-dimensional algebra over a field. An important place among such algebraic structures is occupied by matrix algebras. The impossibility of constructing algebras with division does not at all mean the impossibility of constructing algebras without division, but their properties are close to the first ones (use of defined division). Since each algebra of finite rank can be monomorphically immersed in some complete matrix algebra, this caused, so to speak, an inverse approach to the construction of new algebras. A certain subalgebra stands out from a complete matrix algebra, which is a matrix representation of an algebra of finite rank. It is the implementation of such an approach that makes it possible to endow elements of algebra of finite rank with matrix characteristics, in particular, a canonical representation of algebra elements is constructed through the spectral representation of a matrix, and the algebra itself is endowed with a topological structure through one of the matrix norms. At the same time, an additional condition is often imposed, that it be an algebra over the field of real or complex numbers. The article constructs a real algebra of finite rank, the elements of which are matrices of the second order with the same sum of rows and columns. We endowed it with a norm and a scalar product, demonstrating that it is a Euclidean space. This algebra is a matrix representation of the algebra of hypercomplex numbers, which we called binary in our research [4].
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Takahashi, Kazuhiko. "Comparison of Neural Network–Based Adaptive Controllers Using Hypercomplex Numbers for Controlling Robot Manipulator". IFAC-PapersOnLine 52, nr 29 (2019): 67–72. http://dx.doi.org/10.1016/j.ifacol.2019.12.623.
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Korovin, Aleksandr V., i Ivan V. Aleksandrov. "COORDINATE TRANSFORMATIONS OF THREE-PHASE VARIABLES USING QUATERNIONS". Vestnik Chuvashskogo universiteta, nr 1 (30.03.2022): 65–72. http://dx.doi.org/10.47026/1810-1909-2022-1-65-72.
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Among the variety of modern approaches to the mathematical description of the power quality indicators during the processes of transmission, distribution, conversion and calculation of the ac electric power, the representation of three-phase models in the form of a purely imaginary quaternion located in a separate subspace of the four-dimensional hypercomplex space allows, in relation to the generally accepted method of analyzing linear circuits, for example, symmetrical components with the selection of a direct, reverse and zero phase sequence for the fundamental harmonic, to take into a more complete account the features of energy consumption, especially in the presence of distortion in the modified forms of harmonic signals. In addition, the division of the quaternion into scalar (real) and partial (imaginary) makes it possible to significantly simplify the subsequent analytical processing of synthesis of a power converters control signals for active filtering and power supply of autonomous loads of an arbitrary type, including a single-phase configuration, by extracting from its composition individual components responsible for both the amplitude-phase asymmetry and the nonlinearity of the characteristics.
The main algorithmic principles of organizing control structures as part of three-phase systems of various functional purposes, as a rule, are based on the conversion of reference signals and current values of measured currents and voltages into state coordinates obtained by rotating the three-dimensional space plane by a given angle. At the same time, the calculated ratios for the numerical determination of the initial variables transformed by rotation in the quaternion basis are a function of only four kinematic parameters, which, other things being equal, leads to a simplification of the control law in relation to the traditional vector-matrix approach using nine direction cosines with six connection equations. In this regard, this paper is devoted to the applied problems of implementing linear transformations by E. Clarke and R.H. Park in terms of four-dimensional hypercomplex numbers, in compliance with the additional requirement of the invariance of scalar quantities after the transition.
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Gu, Yingqiu. "Miraculous Hypercomplex Numbers". Mathematics and Systems Science 1, nr 1 (9.11.2023). http://dx.doi.org/10.54517/mss.v1i1.2258.
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<p>The development of number system has gone through a long and difficult process, and many landmark concepts and theorems were put forward. By briefly reviewing the development of hypercomplex systems, the constructing rules of the unit elements are discussed. As a vector space defining multiplication, division and norm of vectors, hypercomplex numbers synthesize the advantages of mathematical tools such as algebra, geometry and analysis, faithfully describe the intrinsic properties of space-time and physical systems, and provide a unified language and a powerful tool for basic theories and engineering technology. In the application of hypercomplex numbers, the group-like properties of the basis vectors are the most important, and the zero factor has little influence on the algebraic operation. The multiplication table of the basis vectors fully describes the intrinsic properties of the hypercomplex system, and the matrix A constructed from the multiplication table satisfies the structure equation A<sup>2</sup> = nA, and thus obtains a set of faithful matrix representations of the basis elements. Th/is paper also uses typical examples to show the simple and clear concepts and wide application of hypercomplex numbers. Therefore, hypercomplex numbers are worth of learning in basic education and appling in scientific research and engineering technology</p>
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"The MIMO Data Transfer Line with Three-Frequency Quaternion Carrier". Journal of Sensor Networks and Data Communications 4, nr 2 (6.05.2024): 01–17. http://dx.doi.org/10.33140/jsndc.04.02.01.
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With the development of communication systems and the Internet, as well as with the growth of services created on the basis of communication networks, the need to increase the capacity of communication channels is increasing. Theoretically, it is shown that it is possible to increase the capacity of communication channels and exceed the “Shannon limit” by moving from the real space of signals on a plane to a multidimensional one with dimension M. In a multidimensional space, each signal is a multidimensional vector, and when such a signal passes through a channel, a MIMO (Many-Input – Many-Output) scheme is formed. As an alternative to existing methods for implementing a MIMO scheme in a physical space with multiple antennas at the input and output of a communication channel, a method for transmitting information using a MIMO scheme in a hypercomplex vector space with one antenna for transmission and one for reception is proposed for wireless communication systems and communication cables. It is known that hypercomplex numbers are an extension of complex numbers through the doubling procedure and form a hypercomplex space on imaginary orthogonal axes and one scalar axis orthogonal to them. For example, a quaternion in algebraic form is written as q= s+xi+yj+zk, where s, x, y, z – real numbers, i, j, k – imaginary units. A quaternion forms a four-dimensional (4D) space. Hypercomplex numbers are also widely known, such as the octonion in 8D and sedenion in 16D spaces. Accordingly, based on these numbers, MIMO schemes with dimensions of 4D, 8D, 16D are implemented. Let us represent the mathematical model of a MIMO channel in a hypercomplex space using a square channel matrix of dimension MxM. From an energy point of view, this MIMO channel model is equivalent to the antenna diversity MIMO model. With an orthogonal channel matrix, maximum capacity is ensured. To synthesize the channel matrix, an exponential function of the quaternion and a polar form of representation of exponentials of imaginary units were used. To get rid of imaginary units in the algebraic form of writing a quaternion and for the purpose of forming a channel matrix, it is represented as a real matrix of dimension 4x4, i.e. three-frequency fundamental matrix Ф(ωi ,ωj .ωk ,t). Using trigonometry formulas, the channel matrix is decomposed into 4 single-frequency matrices of combination frequencies: Ω1 =ωi +ωj +ωk , Ω2 =ωi +ωj -ωk ,, Ω3 =ωi -ωj +ωk ,, Ω4 =ωi -ωj -ωk . A three-frequency channel matrix will, accordingly, be equal to the sum of single-frequency matrices. Modulation of subcarrier frequencies is carried out by multiplying the channel matrix by information vectors: y(ωi ,ωj .ωk ,t)=Ф(ωi ,ωj .ωk ,t)x(0). As a result of multiplication, we obtain QPSK modulation for each combination frequency. When adding frequencies, we obtain a multifrequency oscillation in each element of the modulated vector. Moreover, each element of the output vector y(ωi ,ωj .ωk ,t) contains information about all elements of the information vector x(0), transmitted at all 4 combination frequencies. Elements of the modulated vector are transmitted sequentially as information elements arrive, and the speed of information transmission at the output is equal to the speed of information arrival at the input of the transmitter. Moreover, each multifrequency element consumes the entire transmitter power, which is distributed between 4 frequencies and 4 spatial orthogonal coordinate axes. In addition, in the proposed solution only the elements of the multi-frequency vector are transmitted, and not the elements of the channel matrix, as in existing methods. Thus, using the space-time channel matrix synthesized on the basis of a hypercomplex number, we implement the MIMO scheme in M=2n - dimensional vector space, where n=2,3,4,… for the number of frequencies F=2M-2. According to the transmission model, interference is added to each pulse of the modulated vector when passing through the communication channel. It is clear that the interferences are not correlated, and with a constant dispersion of the interferences, the interference vector has circular symmetry. Consequently, the interference power is distributed along orthogonal axes and frequencies. Since the interference is white noise, the optimal receiver for the received vector will be a correlator using transposed basis matrices for each combinational frequency. By multiplying incoming modulated multi-frequency elements in the sum with interference by single-frequency basis matrices with subsequent integration, we obtain an estimate of each information element with uncorrelated interference and at different frequencies. Since when summing signals add up by energy, and noise - by power, the gain in the signalto-noise ratio (SNR) based on different basic matrices for the quaternion will be equal to 4. After adding up the obtained estimates for different frequencies, we also obtain a gain in SNR by another 4 times. The total gain in SNR will be 16. It is possible to increase the information transmission speed by an appropriate number of times at a given transmission power. By expanding the frequency band, it is also possible to increase the noise immunity and secrecy of transmitter operation. Since the same information is transmitted in each symbol of a multi-frequency vector, the noise immunity to signal fading in time and frequency increases
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Sovetov, V. M. "Quaternion domain Laplace transform of the pulse vector". Electromagnetic Waves and Electronic Systems, 2021. http://dx.doi.org/10.18127/j15604128-202104-03.
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The Laplace transform (LT) is widely used in radio engineering for signal and circuit analysis. The PL greatly facilitates the solution of differential equations, the calculation of transfer functions, the finding of impulse responses, etc. Multiple-Input Multiple-Output (MIMO) systems are becoming more common today. With input influences on such systems, at the output signals are obtained, the elements of which are closely related to each other, and changes in some influencing elements of the input vector change the values of others. Such changes are usually associated with the preservation of the vector norm during transformation. Obviously, this completely changes the shape of the output response and, accordingly, its spectrum. To calculate such changes, it is possible to use the usual PL of real signals and the corresponding theorems. However, this approach requires a significant investment of time and computational resources. If you change the amplitude, shape, time shift of at least one pulse, you will have to repeat all the calculations again. Quaternion transformations, including the Laplace transform, have been studied in many works. However, these studies are often of a general theoretical nature or are used only to obtain the Fractional Quaternion Laplace Transform of 2D images. To calculate the LT of the impulse vector when using the MIMO scheme, it is proposed to use hypercomplex numbers, in the particular case, quaternions. Quaternion is a hypercomplex number with one scalar and three imaginary numbers i, j, k. To get rid of operations with imaginary numbers, the quaternion is represented as an orthogonal 4×4 matrix. The matrix, in turn, is decomposed into 4 basis matrices. Moreover, operations with matrices correspond to operations with imaginary units and the quaternion as a whole. It is shown that the quaternionic Laplace transform (QLT) of the vector is represented as a one-dimensional integral from 0 to ∞ of the vector. In this case, the matrix exponent in the power of the quaternion frequency matrix S = Eσ + 1/√3(I + J + K)ω is used as the transformation kernel, where E, I, J, K are basis matrices. The main properties of the QLT are considered. It is shown that in terms of the notation form, the properties of the QLT correspond to the properties of the LT of real functions, taking into account the non-commutativity of matrix multiplication. Therefore, to calculate the QLT, it is possible to use the well-known expressions for the LT of real pulses with the replacement of the complex frequency s by the matrix of quaternion frequencies S. Expressions for the QLT are obtained for the pulse vectors, which are often used to solve radio engineering problems. It is shown that for σ = 0 these expressions correspond to the quaternionic Fourier transform of the vector pulses. In general, vector pulses can have different delays, amplitudes and shapes. Expressions are obtained for finding the QLT of such vectors.
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"The MIMO Data Transfer Line with Seven-Frequency Octonion Carrier". Engineering: Open Access 2, nr 3 (10.06.2024): 01–23. http://dx.doi.org/10.33140/eoa.02.03.01.
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Currently, with the development of public relations and production systems, there is a need to increase the capacity of communication systems and information transmission. It has been shown theoretically that it is possible to increase throughput by using multidimensional signals in space instead of real signals on a plane. It is now accepted that a multidimensional space, MultipleInput Multiple-Output (MIMO), can be formed using multiple antennas to transmit and receive in physical space. However, as physicists point out, such space is three-dimensional, and with the addition of time it is four-dimensional. It is clear that in such a physical space, when using more than 2 antennas for transmission and 2 for reception, it is impossible to obtain a gain in throughput of more than 4 times, since according to the laws of cybernetics, the diversity at the channel input will not be transmitted to the exit. It follows that it is necessary to reconsider existing views on the dimension of physical space. Previously, in the work the MIMO data transfer line with three-frequency quaternion carrier, it was shown that it is possible to use a hypercomplex quaternion number as a model of physical space. In this case, the dimension of space will be equal to 4 with 3 imaginary (spatial) axes and one scalar axis. In addition, combinations of three quaternion angular frequencies on the imaginary axes formed 4 single-frequency channels. Accordingly, the gain in throughput compared to real signals reached 4 in orthogonal axes and 4 in frequencies. In this work, an octonion with 7 imaginary (spatial) axes and one scalar is used as a mathematical model of physical space. It is shown that the dimension of physical space will be 8 with 64 single-frequency channels in the form of combinations of 7 angular frequencies. Hence, the gain in throughput will be 8 in orthogonal axes and 64 in frequencies.
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Belenok, N. V., V. I. Chibelis i L. Yu Spinul. "BICOMPLEX ANALYSIS OF INVARIENT POWER SUPPLY SYSTEMS BASED ON RENEWABLE ENERGY SOURCES". POWER ENGINEERING: economics, technique, ecology, nr 2 (14.07.2022). http://dx.doi.org/10.20535/1813-5420.2.2022.261371.
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The article considers a bicomplex calculation for calculating the invariant power supply systems based on renewable energy sources. Modern energy supply systems based on renewable energy sources have non-linear systems with complex transients and possible critical and chaotic regimes. The study of structures of hypernumerical systems, their features, methods of calculation and approximation of the elementary functions of a hypercomplex variable allows to effectively apply such systems in mathematical modelling of invariant power supply systems based on renewable energy sources. In some cases, the use of hypernumerical systems makes it possible to replace the original problem with an equivalent one, that is to build a bicomplex solution model.
The system of complex numbers was considered as the initial system. With recurrent doubling of the system, hypernumerical systems of different dimensions with different properties were obtained, which made it possible to assign different values to the products of imaginary units. It is proved that the introduction of additional conditions of commutativity and associativity, which apply to real numbers and imaginary units, allows to specify the choice of a hypernumerical system.
In the analysis of nonstationary processes of invariant systems and the study of the possibilities of hypernumerical systems, the expediency of choosing a bicomplex calculation method in mathematical modelling of systems with multiple modulation is substantiated. The method of bicomplex representation involves direct and inverse bicomplex transformation, which allows obtaining an analytically complete solution for the analysis of an invariant power supply system based on renewable energy sources. Examples of the use of bicomplex integral transformation for the analysis of systems with multiple modulation are considered. The application of the hypercomplex calculus apparatus for the transformation of systems of differential equations is proposed to simplify or compress them into one equation. It is shown that the use of hypercomplex calculus allows to significantly reduce the amount of processed information without reducing the informativeness of the mathematical model.
The proposed formulation of tasks in a hypercomplex view allowed to compress the processing information and obtain a compact vortex for the output signal.
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Tsiotras, Panagiotis, Matthew King-Smith i Lorenzo Ticozzi. "Spacecraft-Mounted Robotics". Annual Review of Control, Robotics, and Autonomous Systems 6, nr 1 (7.10.2022). http://dx.doi.org/10.1146/annurev-control-062122-082114.
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Space-mounted robotics is becoming increasingly mainstream for many space missions. The aim of this article is threefold: first, to give a broad and quick overview of the importance of spacecraft-mounted robotics for future in-orbit servicing missions; second, to review the basic current approaches for modeling and control of spacecraft-mounted robotic systems; and third, to introduce some new developments in terms of modeling and control of spacecraft-mounted robotic manipulators using the language of hypercomplex numbers (dual quaternions). Some outstanding research questions and potential future directions in the field are also discussed. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 14 is May 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
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Błaszczyk, Łukasz. "Discrete octonion Fourier transform and the analysis of discrete 3-D data". Computational and Applied Mathematics 39, nr 4 (22.11.2020). http://dx.doi.org/10.1007/s40314-020-01373-7.
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AbstractThe purpose of this article is to further develop the theory of octonion Fourier transformations (OFT), but from a different perspective than before. It follows the earlier work by Błaszczyk and Snopek, where they proved a few essential properties of the OFT of real-valued functions of three continuous variables. The research described in this article applies to discrete transformations, i.e. discrete-space octonion Fourier transform (DSOFT) and discrete octonion Fourier transform (DOFT). The described results combine the theory of Fourier transform with the analysis of solutions for difference equations, using for this purpose previous research on algebra of quadruple-complex numbers. This hypercomplex generalization of the discrete Fourier transformation provides an excellent tool for the analysis of 3-D discrete linear time-invariant (LTI) systems and 3-D discrete data.